Question

What is the average speed of an object that is moving at `4 m/s` at `t=0` and accelerates at a rate of `a(t) =2-t` on `t in [0,3]`?

Answer 1

`5.5 m/s`

Explanation:

I'm assuming this is related to one-dimensional motion. Combining the integral functions for velocity and position, the equation for position with respect to time is represented by

`x = x_0 + int_0^t [v_(0x) + int_0^ta_xdt]dt`

So, since the initial velocity `v_(0x) = 4 m/s`, the postion equation with respect to time from this is

`x = 4m/s(t) + 1m/(s^2)(t)^2 - 1/6m/(s^3)(t)^3`

and thus the position of the object at time `t = 3` is

`x = 4m/s(3) + 1m/(s^2)(3)^2 - 1/6m/(s^3)(3)^3 = 16.5m`

The average velocity on the interval `t in [0,3]` is thus

`v_(av-x) = (Deltax)/(Deltat) = (16.5m)/(3s) = 5.5 m/s`